Restriction of Averaging Operators to Algebraic Varieties over Finite Fields

Abstract

We study $L^p \to L^r$ estimates for restricted averaging operators related to algebraic varieties $V$ of $d$-dimensional vector spaces over finite fields $\mathbb{F}_q$ with $q$ elements. We observe properties of both the Fourier restriction operator and the averaging operator over $V \subset \mathbb{F}_q^d$. As a consequence, we obtain optimal results on the restricted averaging problems for spheres and paraboloids in dimensions $d \geq 2$, and cones in odd dimensions $d \geq 3$. In addition, when the variety $V$ is a cone lying in an even dimensional vector space over $\mathbb{F}_q$ and $-1$ is a square number in $\mathbb{F}_q$, we also obtain sharp estimates except for two endpoints.